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Periodicity of Non-Central Integral Arrangements Modulo Positive Integers
Aaron D. Jaggard1 and Joseph J2. Marincel3
1Graduate School of Economics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8601, Japan
kamiya@soec.nagoya-u.ac.jp
2Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan
takemura@stat.t.u-tokyo.ac.jp
3Department of Mathematics, Hokkaido University, North 10, West 8, Kita-ku, Sapporo, 060-0810, Japan
hterao00@za3.so-net.ne.jp
Annals of Combinatorics 15 (3) pp.449-464 July, 2011
AMS Subject Classification: 32S22; 52C35
Abstract:
An integral coefficient matrix determines an integral arrangement of hyperplanes in Rm. After modulo q reduction (q ∈ Z>0), the same matrix determines an rrangement
Aq of “hyperplanes” in Zmq . In the special case of central arrangements, Kamiya, Takemura, and Terao [J. Algebraic Combin. 27(3), 317–330 (2008)] showed that the cardinality of the complement of Aq in Zmq is a quasi-polynomial in q ∈ Z>0. Moreover, they proved in the central case that the intersection lattice of Aq is periodic from some q on. The present paper generalizes these results to the case of non-central arrangements. The paper also studies the arrangement ˆB [0,a]
m of Athanasiadis [J. Algebraic Combin. 10(3), 207–225 (1999)] to illustrate our results.
Keywords: characteristic quasi-polynomial, elementary divisor, hyperplane arrangement, intersection poset

References:

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